Mohamed Elhamdadi

University of South Florida | USF · Department of Mathematics & Statistics

We completely characterize the coloring quivers of general torus links, T(p, q) where p is prime, by dihedral quandles by first exhausting all possible numbers of colorings, followed by determining the interconnections between colorings in each case. The quiver is obtained as a function of the number of colorings. The quiver always contains complet...

This article fully characterize the residual finiteness of conjugation quandles of the Baumslag-Solitar groups. The key difference between groups and their derived conjugation quandles is that many conjugation quandles are infinitely generated even when their underlying groups are finitely generated. Notably, we prove that the conjugation quandles...

We introduce a generalization of the quandle polynomial. We prove that our polynomial is an invariant of stuquandles. Furthermore, we use the invariant of stuquandles to define a polynomial invariant of stuck links. As a byproduct, we obtain a polynomial invariant of RNA foldings. Lastly, we provide explicit computations of our polynomial invariant...

While knotoids on the sphere are well-understood by a variety of invariants, knotoids on the plane have proven more subtle to classify due to their multitude over knotoids on the sphere and a lack of invariants that detect a diagram's planar nature. In this paper, we investigate equivalence of planar knotoids using quandle colorings and cocycle inv...

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of the notion of left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebras from O-operators on Lie-Rinehart algebras. We extensively investig...

This paper gives a new way of characterizing L-space 3-manifolds by using orderability of quandles. Hence, this answers a question of Clay et al. (Question 1.1 of Can Math Bull 59(3):472–482, 2016). We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the c...

This article serves two purposes. The first is to give an introduction to the readers who are not so familiar with quandle theory. The second is to report on new results. The first new result is that we compute second and third homology groups of disjoint union of quandles using the machinery of spectral sequences. We also establish a connection be...

We introduce and investigate dichromatic singular links. We also construct disingquandles and use them to define counting invariants for unoriented dichromatic singular links. We provide some examples to show that these invariants distinguish some dichromatic singular links.

We introduce the notion of relative averaging operators on Hom-associative algebras with a representation. Relative averaging operators are twisted generalizations of relative averaging operators on associative algebras. We give two characterizations of relative averaging operators of Hom-associative algebras via graphs and Nijenhuis operators. A (...

This paper gives a new way of characterizing L-space $3$-manifolds by using orderability of quandles. Hence, this answers a question of Adam Clay et al. [Question 1.1 of Canad. Math. Bull. 59 (2016), no. 3, 472-482]. We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions...

Quandle representations are homomorphisms from a quandle to the group of invertible matrices on some vector space taken with the conjugation operation. We study certain families of quandle representations. More specifically, we introduce the notion of regular representation for quandles, investigating in detail the regular representations of dihedr...

An Alexandroff space is a topological space in which every intersection of open sets is open. There is one to one correspondence between Alexandroff T0 -spaces and partially ordered sets (posets). We investigate Alexandroff T0 -topologies on finite quandles. We prove that there is a non-trivial topology on a finite quandle making right multiplicati...

We extend the quandle cocycle invariant to the context of stuck links. More precisely, we define an invariant of stuck links by assigning Boltzmann weights at both classical and stuck crossings. As an application, we define a single-variable and a two-variable polynomial invariant of stuck links. Furthermore, we define a single-variable and two-var...

We investigate finite right-distributive binary algebraic structures called shelves. We first use symbolic computations with Python to classify (up to isomorphism) all connected shelves with order less than six. We explore the group structure generated by the rows of latin shelves. We also define two-variable shelf polynomial by analogy with the qu...

In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are nontrivial coverings over nice base quandles admit infinitely many nontrivial idempotents, and give their complete description. We show that the set of all these idempotents form...

We introduce and investigate dichromatic singular links. We also construct G-Family of singquandles and use them to define counting invariants for unoriented dichromatic singular links. We provide some examples to show that these invariants distinguish some dichromatic singular links.

We introduce a notion of ternary $F$-manifold algebras which is a generalization of $F$-manifold algebras. We study representation theory of ternary $F$-manifold algebras. In particular, we introduce a notion of dual representation which requires additional conditions similar to the binary case. We then establish a notion of a coherence ternary $F$...

We extend the quandle cocycle invariant to the context of stuck links. More precisely, we define an invariant of stuck links by assigning Boltzmann weights at both classical and stuck crossings. As an application, we define a single-variable and a two-variable polynomial invariant of stuck links. Furthermore, we define a single-variable and two-var...

An Alexandroff space is a topological space in which every intersection of open sets is open. There is one to one correspondence between Alexandroff $T_0$-spaces and partially ordered sets (posets). We investigate Alexandroff $T_0$-topologies on finite quandles. We prove that there is a non-trivial topology on a finite quandle making right multipli...

We introduce a cohomology theory of $n$-ary self-distributive objects in the tensor category of vector spaces that classifies their infinitesimal deformations. For $n$-ary self-distributive objects obtained from $n$-Lie algebras we show that ($n$-ary) Lie cohomology naturally injects in the self-distributive cohomology and we prove, under mild addi...

We study RNA foldings and investigate their topology using a combination of knot theory and embedded rigid vertex graphs. Knot theory has been helpful in modeling biomolecules, but classical knots place emphasis on a biomolecule's entanglement while ignoring their intrachain interactions. We remedy this by using stuck knots and links, which provide...

We show that quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space. We give explicit examples to show that the new invariants are also stronger than the $\Hom$ quandle invariant when the coloring quandles are medial. We provide computer assisted computations of idempoten...

We introduce a notion of n-Lie–Rinehart algebras as a generalization of Lie–Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie–Rinehart algebras. Furthermore, we investigate extension theory of n-Lie–Rinehart algebras by means of...

We decompose the regular quandle representation of a dihedral quandle $\mathcal{R}_n$ into irreducible quandle subrepresentations.

We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely g...

The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fi...

In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are non-trivial coverings over nice base quandles admit infinitely many non-trivial idempotents, and give their complete description. We show that the set of all these idempotents fo...

In this paper, we introduce the notion of circular orderability for quandles. We show that the set all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right (respectively left) orderings of a quandle embeds in its space of right (respectively left) circular orderings. Examples...

In this short survey, we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polyn...

We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called oriented singquandles and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots, but provides extra information about singular knots and links....

We introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial in...

We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB operator R for Jones, normalized for homology, admits a skein decomposition R=I+βα, where α:V⊗2→k is a “cup” pairing map and β:k→V⊗2 is a “cap” copairing...

Folded linear molecular chains are ubiquitous in biology. Folding is mediated by intra-chain interactions that “glue” two or more regions of a chain. The resulting fold topology is widely believed to be a determinant of biomolecular properties and function. Recently, knot theory has been extended to describe the topology of folded linear chains, su...

The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of \emph{dihedral quandles}...

Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a folded linear chain, such as a protein molecule or the genome. Theses linear biomolecular chains often fold into complex 3D architectures with critical entanglements and local or global structural symmetries stabilised by formation of intrachain contac...

We introduce a notion of $n$-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to $n$-ary case. This notion is also an algebraic analogue of $n$-Lie algebroids. We develop representation theory and describe a cohomology complex of $n$-Lie Rinehart algebras. Furthermore, we investigate extension theory of $n$-Lie Rinehart algebras b...

In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polyno...

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, usin...

Heaps are algebraic structures endowed with para-associative ternary operations, bijectively exemplified by groups via the operation ( x , y , z ) ↦ x y − 1 z . They are also ternary self-distributive and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theori...

We introduce shadow structures for singular knot theory. Precisely, we define \emph{two} invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polyno...

We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram.

Folded linear molecular chains are ubiquitous in biology. Folding is mediated by intra-chain interactions that "glue" two or more regions of a chain. The resulting fold topology is widely believed to be a determinant of biomolecular properties and function. Recently, knot theory has been extended to describe the topology of folded linear chains suc...

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, usin...

The tail of a quantum spin network in the two-sphere is a [Formula: see text]-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by [Formula: see text]. We compute the [Formula: see text]-series for an infinite family of quantum spin networks and give the relation between the tai...

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from $\mathcal O$-operators on Lie-Rinehart algebra. We extensively investigate...

We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called \emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and l...

Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that 97% of proteins are, however, classified as unknots when intra-chain interactions are ignored. This raises the...

This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled...

We investigate generalized derivations of n-BiHom-Lie algebras. We introduce and study properties of derivations, \(( \alpha ^{s},\beta ^{r}) \)-derivations and generalized derivations. We also study quasiderivations of n-BiHom-Lie algebras. Generalized derivations of \((n+1)\)-BiHom-Lie algebras induced by n-BiHom-Lie algebras are also considered.

We use the structural aspects of the f-quandle theory to classify, up to isomorphisms, all f-quandles of order n. The classification is based on an effective algorithm that generate and check all f-quandles for a given order. We also include a pseudocode of the algorithm.

We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram.

We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB operator $R$ for Jones, normalized for homology, admits a skein decomposition $R = I + \beta\alpha$, where $\alpha: V^{\otimes 2} \rightarrow k$ is a "cu...

Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that 97% of proteins are, however, classified as unknots when intra-chain interactions are ignored. This raises the...

This article gives a foundational account of various characterizations of framed links in the $3$-sphere

This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.

Heaps are para-associative ternary operations bijectively exemplified by groups via the operation $(x,y,z) \mapsto x y^{-1} z$. They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories and also a ternary self-distr...

We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian st...

We investigate constructions and relations of higher arity self-distributive operations and their cohomology. We study the categories of mutually distributive structures both in the binary and ternary settings and their connections through functors. This theory is also investigated in the context of symmetric monoidal categories. Examples from Lie...

We investigate generalized derivations of $n$-BiHom-Lie algebras. We introduce and study properties of derivations, $( \alpha^{s},\beta^{r}) $-derivations and generalized derivations. We also study quasiderivations of $n$-BiHom-Lie algebras. Generalized derivations of $(n+1)$-BiHom-Lie algebras induced by $ n $-BiHom-Lie algebras are also considere...

The tail of a quantum spin network in the two-sphere is a $q$-series associated to the network. We study the existence of the head and tail functions of such families in the two-sphere. We compute the $q$-series for a family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomia...

This is an Erratum for our paper “Singular Knots and Involutive Quandles” in [1]. It replaces Sec. 5 in the published version of the paper.

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket s...

We associate to every quandle X and an associative ring with unity k, a nonas-sociative ring k[X] following Bardakov et al. (Quandle rings, arXiv:1709.03069, [3]). The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group Inn(X) acts orbit 2-transitively on X, a complete description o...

We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a \textit{nonassociative} ring $\mathbf{k}[X]$ following \cite{BPS}. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group $Inn(X)$ acts \textit{orbit $2$-transitively} on $X$, a complete descriptio...

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the invers...

We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval $[0, 1]$, we conjecture that there exists only one topological quandle structure on it, i.e. the trivial one. Some evidences...

We introduce and study ternary $f$-distributive structures, Ternary $f$-quandles and more generally their higher $n$-ary analogues. A classification of ternary $f$-quandles is provided in low dimensions. Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally $n$-ary, $f$-quandles. Furthermore, we give...

We determine the second, third, and fourth cohomology groups of Alexander $f$-quandles of the form $\mathbb{F}_q[T,S]/ (T-\omega, S-\beta)$, where $\f_q$ denotes the finite field of order $q$, $\omega \in \f_q\setminus \{0,1\}$, and $\beta \in \f_q$.

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and co...

We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by...

We study derivations of ternary Lie algebras. Precisely, we investigate the relation between derivations of Lie algebras and the induced ternary Lie algebras. We also explore the spaces of quasi-derivations, the centroid and the quasi-centroid and give some properties. Finally, we compute these spaces for low dimensional ternary Lie algebras g.

We investigate the colorings of knots by the linear Alexander quandles of order five. Precisely, we prove that any colorable knot by a linear Alexander quandle of order five has a diagram using only four colors of the five colors that are assigned to the arcs of the diagram.

The purpose of this paper is to introduce and study the notions of f-rack and f-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order f-quandles. Moreover, we define modules over f-racks, discuss extensions and define a cohomology theory for f-qua...

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavours with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove...

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.

The purpose of this paper is to introduce and study the notions of twisted rack and twisted-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order twisted-quandles. Moreover, we define modules over twisted-racks, discuss extensions and define a coh...

We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $q$-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.

The Kauffman-Vogel polynomial is an invariant of unoriented four-valent graphs embedded in three dimensional space. In this article we give a one variable generalization of this invariant. This generalization is given as a sequence of invariants in which the first term is the Kauffman-Vogel polynomial. We use the invariant we construct to give a se...

We generalize the colored Jones polynomial to 4-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.